Bricks and pseudo MV-algebras are equivalent

Abstract

ABSTRACT. We will show that the bricks (of Bosbach) and the pseudo MV-algebras are each term equivalent to the class of semigroups with a pair of unary operationsˆandˇsatisfying the equations: (âa)ˆb = b = b (aǎ)ˇand a(ba)ˇ= (bǎ)ˆb and also show that a brick is an interval [0, u] of the positive cone of a unital lattice ordered group. We further extend the notion of implications to a pseudo MV-algebra and study the algebra of such implications.

Commutative extended BCK-algebras

“…Hence if (C; ≤, * , ∶ , e) is a commutative extended BCK-algebra bounded above, then C is a commutative pseudo BCK-algebra bounded above and hence, by Theorem 1.11, is a distributive lattice. Further, by Corollary [10, 5.8], C is a bounded precone algebra and hence is a brick ( [2,12]) and hence is equivalent to a pseudo MV-algebra ( [11]). See Theorem [6, 3.14].…”

A Note on Bosbach’s Cone Algebras

“…In the same article, he introduced his concept of brick which can be regarded as a cone algebra with smallest element 0 (see section 2). Therefore, an alternative proof of Dvurečenskij's result [5] can be obtained by showing that bricks are equivalent to pseudo-MV algebras (see [15,16]). For an extension to non-integral GMV-algebras in the sense of Galatos and Tsinakis, see [6].…”

Deductive systems of a cone algebra — I: Semi-ℓg-cones

“…We also need the following lemma from [12], which is a simple consequence of the above Definition 3.9. (Also, see [4].)…”

“…R a c hů n e k [10], under the name "generalized MV-algebra". We have shown recently ( [12]) that a pseudo MV-algebra is term equivalent to a brick; and we show in this part, in a more general context, that a subset A of a pseudo MV-algebra C is an ideal of C ( [11, p. 156…”